Linda writes:
"I have never heard of a breeder producing miniature or teacup corgis."
I have.

Great marketing scam to rip people off, but I've seen them advertised. Price tag $1,500 plus. Any breeder advertising such, my advice would be to run not walk away as fast as one could.
Behindhereyes is exactly right though, there is no such thing. There are undersized corgis though and I know several. I've had a very petite girl a volume less than scrupulous breeder sold off because she wasn't able to carry litters to term (GEE, I wonder why?) and there was no money to be made off of her. She was overweight at 19 LBS, 17 to 18 LBS were the max she needed to carry. I know several adults that weigh in at 20 LBS or a little less. They are NOT teacup corgis, just undersized and small and they happen.
My recommendation would be to check with breeders in your area who might tend to have smaller corgis. <http://www.pwcca.org> under the members/breeders link will be a good place to start. And even though overall someone's corgis may tend to run on the lower end of the standard, doesn't mean that when they are full grown they won't grow into the middle or higher end of the scale. I know someone who had grand hopes for a 10" corgi for agility so they could run in the 8 inch class. This owner worked with a very responsible Breeder known for her correctly moderate corgis, some even leaning towards the smaller side. Puppies were evaluated for structure, temperament, size for the best potential candidate. Well those hopes were dashed when she reached 11" in height and now has to jump in the 12 inch class. She's flying, just a little higher over the clouds than was wished for. The message here is Mother Nature will have the last word regardless.
Debbie